ABEL'S THEOREM AND ABELIAN FUNCTIONS. 
But a x , a 2 , . . . a p are the p roots of 
P(z)=0 
and therefore as before 
and so 
and therefore 
=1 |F(a») 
U+V+W=-N x 
m =P n P- 1 m =P n p-s 
£f k) =land i,ffe =o(s>1 ^ 
m=p r f 
t 1 ^,; l --al m (u)td m (v)al m (u+v) 1 = ±N 1 
w=p r i 
= $ 1B77 l —,al M (u)cd m (v)cd m (u+v) 
m= 1 L-*- 
On the expansion of each side in terms of the u ’s and v's as is done below, it is at 
once seen that the lower sign is the correct one; and therefore 
m=p r 7 
U+Y+W= 2 al m (u)al m (v)al m (u+v) 
m=l L 1 \ a m) 
This may be called the addition theorem for the integral-function ; by putting 
p= 1 and referring to the example worked out in the last section, it is at once seen 
to be the addition theorem for elliptic integrals of the second order. 
22. In the expansion of the two sides in terms of u ’s and vs the first term is 
sufficient to indicate the correct sign in the above; but it is not uninteresting to see 
the agreement for terms of a higher order, and the expansion is carried on as far as 
the order seven in the magnitudes u. 
Proceeding therefore to form the expansion of U in terms of the u’ s, write, with 
Weierstrass, 
PVY 
lr 
{cL r —x r )—s? .(24) 
so that 
and 
Let 
so that 
— — ( a ^ dx r — 2 s r ds r 
L r 
d.Vr 
x r —a r s r 
7 tl T)// \ 
P ( a r ) a r —a„, 
Ijn ly 
PW*"- 
(25) 
then 
